The Leggett-Garg inequality of a dissipative cavity mode coupled to a zero-temperature environment
Hiroo Azuma, Masashi Ban · arXiv (Cornell University) · 2021
In the present paper, we investigate the Leggett-Garg inequality of the cavity mode weakly coupled to a zero-temperature environment. We assume that the boson system undergoes dissipation because of an interaction with the environment but is not affected by dephasing. Solving the master equation exactly, we derive an explicit form of violation of the inequality for both cases where systems are prepared initially in the coherent state $|\alpha\rangle$ and the cat state $(|\alpha\rangle+|-\alpha\rangle)$. We choose the displaced parity operators for observing the cavity mode at three equally spaced measurement times (spacing $\tau$) in the derivation of the inequality. These operators are characterized by a complex number $\beta$. We look for the optimum parameter $\beta$ that lets the upper bound of the inequality be maximum by numerical calculations. We compare the maximized upper bounds for both the initial states, the coherent and cat states. Contrary to our expectations, the coherent state occasionally exhibits quantum quality more strongly than the cat state for the upper bound of the Leggett-Garg inequality if its time difference $\tau$ is in a specified range. Moreover, as we let the time difference $\tau$ approach zero, the optimized parameter $\beta$ diverges, that is $|\beta|\to\infty$, and the Leggett-Garg inequality reveals intense singularity.